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Chapter 5 Ramanujan's second notebook

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Analytic Number Theory

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References

  1. M. Abramowitz and I.A. Stegun, editors, Handbook of mathematical functions, Dover, New York, 1965.

    Google Scholar 

  2. G.E. Andrews, An incredible formula of Ramanujan, Australian Math. Soc. Gazette 6 (1979), 80–89.

    MathSciNet  MATH  Google Scholar 

  3. P. Appell, Sur une classe de fonctions analogues aux fonctions Eulériennes, Math. Ann. 19 (1882), 84–102.

    Article  MathSciNet  Google Scholar 

  4. R. Ayoub, An introduction to the analytic theory of numbers, American Mathematical Society, Providence, 1963.

    MATH  Google Scholar 

  5. R. Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067–1086.

    Article  MathSciNet  MATH  Google Scholar 

  6. B.C. Berndt, Elementary evaluation of ξ(2n), Math. Mag. 48 (1975), 148–154.

    Article  MathSciNet  MATH  Google Scholar 

  7. B.C. Berndt, P.T. Joshi, and B.M. Wilson, Chapter 2 of Ramanujan's second notebook, Glasgow Math. J. (to appear).

    Google Scholar 

  8. B.C. Berndt and L. Schoenfeld, Periodic analogues of the Euler-Maclaurin and Poisson summation formulas with applications to number theory, Acta Arith. 28 (1975), 23–68.

    MathSciNet  MATH  Google Scholar 

  9. T.B.N. Brodén, Bemerkungen über sogenannte finite Integration, Arkiv för Mat. Astr. och Fys. (Stockholm) 7 (1911), 34pp.

    Google Scholar 

  10. T.B.N. Brodén, Einige Anwendungen diskontinuierlicher Integrale auf Fragen der Differenzenrechnung, Acta Univ. Lund. (2) 8 (1912), 17pp.

    Google Scholar 

  11. T.J. I'A. Bromwich, An introduction to the theory of infinite series, second ed., Macmillan, London, 1926.

    MATH  Google Scholar 

  12. L. Carlitz, Eulerian numbers and polynomials, Math. Mag. 33 (1959), 247–260.

    Article  MathSciNet  MATH  Google Scholar 

  13. L. Carlitz, Eulerian numbers and operators, Coll. Math. 24 (1973), 175–200.

    MathSciNet  MATH  Google Scholar 

  14. L. Carlitz, Some remarks on the Eulerian function, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 611, 1978, 79–91.

    MathSciNet  MATH  Google Scholar 

  15. L. Carlitz, D.C. Kurtz, R. Scoville, and O.P. Stackelberg, Asymptotic properties of Eulerian numbers, Z. Wahrscheinlichkeitstheorie 23 (1972), 47–54.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Carlitz and J. Riordan, Congruences for Eulerian numbers, Duke Math. J. 20 (1953), 339–343.

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. Reine Angew. Math. 265 (1974), 110–137 Corrigendum 288 (1976), 218–219.

    MathSciNet  MATH  Google Scholar 

  18. L. Carlitz and R. Scoville, Eulerian numbers and operators, Fibonacci Quart. 13 (1975), 71–83.

    MathSciNet  MATH  Google Scholar 

  19. L. Euler, Institutiones calculi differentialis, Acad. Imperialis Sci., Petrograd, 1755.

    Google Scholar 

  20. F.G. Frobenius, Über die Bernoullischen Zahlen und die Eulerschen Polynome, Sitz. d. K. Preuss. Akad. Wiss. Berlin 1910, 809–847.

    Google Scholar 

  21. F.G. Frobenius, Gesammelte Abhandlungen, Band III, Springer-Verlag, Berlin, 1968.

    Book  Google Scholar 

  22. E. Landau, Primzahlen, Chelsea, New York, 1953.

    MATH  Google Scholar 

  23. A.M. Legendre, Traité des fonctions elliptiques et des intégrales Eulériennes, tome II, Huzard-Courcier, Paris, 1826.

    Google Scholar 

  24. N. Nielsen, Traité élémentaire des nombres de Bernoulli, Gauthier-Villars, Paris, 1923.

    MATH  Google Scholar 

  25. N.E. Nörlund, IIC7. Neure Untersuchungen über Differenzengleichungen, Encyklopädie der mathematischen Wissenschaften, Band 2, Teil 3, B.G. Teubner, Leipzig, 1923, pp. 675–721.

    Google Scholar 

  26. N.E. Nörlund, Vorlesungen über Differenzenrechnung, Chelsea, New York, 1954.

    Google Scholar 

  27. E. Picard, Sur une classe de transcendantes nouvelles, Acta Math. 18 (1894), 133–154.

    Article  MathSciNet  Google Scholar 

  28. S. Ramanujan, Some properties of Bernoulli's numbers, J. Indian Math. Soc. 3 (1911), 219–234.

    MATH  Google Scholar 

  29. S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913), 105–106.

    MATH  Google Scholar 

  30. S. Ramanujan, Collected papers, Chelsea, New York, 1962.

    Google Scholar 

  31. S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.

    MATH  Google Scholar 

  32. J. Riordan, An introduction to combinatorial analysis, John Wiley, New York, 1958.

    MATH  Google Scholar 

  33. J. Riordan, Combinatorial identities, John Wiley, New York, 1968.

    MATH  Google Scholar 

  34. T. J. Stieltjes, Table des valeurs des sommes \(S_k = \sum\limits_1^\infty {n^{ - k} }\), Acta Math. 10 (1887), 299–302.

    Article  MathSciNet  Google Scholar 

  35. J. V. Uspensky and M. A. Heaslet, Elementary number theory, McGraw-Hill, New York, 1939.

    MATH  Google Scholar 

  36. S. S. Wagstaff, Jr., The irregular primes to 125000, Math. Comp. 32 (1978), 583–591.

    MathSciNet  MATH  Google Scholar 

  37. S. S. Wagstaff, Jr., Ramanujan's paper on Bernoulli numbers, submitted for publication.

    Google Scholar 

  38. J. Worpitzky, Studien über die Bernoullischen und Eulerschen Zahlen, J. Reine Angew. Math. 94 (1883), 203–232.

    MathSciNet  Google Scholar 

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Authors

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Marvin I. Knopp

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Dedicated to Emil Grosswald, with respect and admiration.

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© 1981 Springer-Verlag

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Berndt, B.C., Wilson, B.M. (1981). Chapter 5 Ramanujan's second notebook. In: Knopp, M.I. (eds) Analytic Number Theory. Lecture Notes in Mathematics, vol 899. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096453

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  • DOI: https://doi.org/10.1007/BFb0096453

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